170 
SIR G. H. DARWIN ON THE 
§ 3. On the Possibility of joining Tiro Masses of Liquid by a thin Neck. 
This whole investigation was undertaken principally in the hope that it might lead 
to an approximation to the form of the pear-shaped figure of equilibrium of a rotating 
mass of liquid at the stage when it should resemble an hour-glass with a thin neck. 
It seemed probable that such an approximation might be obtained in the following- 
manner :— 
Two masses of liquid are revolving in an orbit about one another without relative 
motion of their parts, so that they form a figure of equilibrium. Imagine them to be 
joined by a pipe without weight, through which liquid may flow from one part 
to the other. A flow of liquid will in general take place between the two parts, but 
there should be some definite partition of masses, corresponding to a given distance 
apart, at which flow will cease. At this stage we should have an approximation 
to the hour-glass figure of equilibrium. 
In this section a special case of this problem is considered, in which the detached 
masses, to be joined by a pipe, are constrained to be spheres. 
If the notation of § 1 be adopted, it is clear that the system is defined by the two 
parameters r and X. In accordance with the notation of § 2 we denote the lost energy 
of the system by V and the moment of inertia by I. It is easily shown that 
For brevity write 
V = (|^)V 
I = #— 15 
\ TTpiX 
X a 3 1 + X 5/3 
jlfNfr 5 
X r 2 l+X 5/3 ~ 
+ - 
L(l+X) 2 a 2 5 (I + X) 5/3 J * 
F = 
X 
(1+X) 
2 ’ 
G = 
1 + X 
5/3 
(1+Xf 3 ’ 
and let F', G', F", G" denote their first and second differentials with respect to X. 
The equations for determining the configuration of equilibrium are 
3F,! 2 a i _ 
+ = °- 
SF . n 
The first of these gives at once 
9 
OJ = 
For determining the form of the second we have 
